Please help with dimensional analysis?

Question

If the length of the pendulum is L and the acceleration of gravity is g, then T is given by T = 2 * pi * L^p * g^q . Find the powers p and q required for dimensional consistency.

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I don't know how to go about this...any help would be great!

Answer 1

T is period measured in seconds.

g is acceleration due to gravity measured in m/s^2.

L is pendulum length measured in meters.

pi is dimensionless.

To find p and q set the dimensions of g and L equal to that of T.

[s]=[m]^p * [m/s^2]^q

Choose on dimension and solve for its exponents. Start with s.

On the left side of the equation, s is not raised to any power, so its exponent is 1. On the right side of the equation, s is squared in the denominator so it has an exponent of -2 which must be multiplied by q, per the law of exponents. Setting them equal gives the following equation.

1= -2*q. Solve and find that q = -1/2.

Do the same for m. On the left, m is not there so its exponent is 0. On the right m is raised once to the p power and once to the q power. Per the law of exponents, these must be added. The resulting equation is:

0 = p +q. Solve and find that p=1/2.

You should find that if you look up the formula for period of a simple pendulum, it will be

T = 2 Pi sqrt(L/g), which matches the answer.